hacer lugar a la medida

English translation: Would have granted the petition requested

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I'm assuming this is in a submission to a court. My guess is that the representatives of a party submitted a request for the court to take some action ("measure"). From the context I don't know what kind of measure it was. It may have been, for example, a "medida cautelar" such as an attachment, freezing a bank account, whatever.

The court initially denied the request. The party wasn't satisfied, is now filing some sort of appeal, has stated some relevant facts ("condiciones"), and wants the court to reconsider, and approve or take the action (measure) that was originally requested.